Interference, fluctuation, and alternation of electron tunneling in protein media. 2. Non-condon theory for the energy gap dependence of electron transfer rate

Developing the quantum transition rate theory of Prezhdo and Rossky (J. Chem. Phys. 1997, 107, 5863), we produced a new non-Condon theory of the rate of electron transfer (ET) which happens through a protein medium with conformational fluctuation. The new theory is expressed by a convolution form of the power spectrum for the autocorrelation function of the electronic tunneling matrix element T(DA)(t) with quantum correction and the ordinary Franck-Condon factor. The new theory satisfies the detailed balance condition for the forward and backward ET rates. The ET rate formula is divided into two terms of elastic and inelastic tunneling mechanisms on the mathematical basis. The present theory is applied to the ET from Bph(-) to Q(A) in the reaction center of Rhodobacter sphaeroides. Numerical calculations of T(DA)(t) were made by a combined method of molecular dynamics simulations and quantum chemistry calculations. We showed that the normalized autocorrelation function of T(DA)(t) is almost expressed by exponential forms. The calculated energy gap law of the ET rate is nearly Marcus' parabola in most of the normal region and around the maximum region, but it does not decay substantially in the inverted region, which is called the anomalous inverted region. We also showed that the energy gap law at the high uphill energy gap in the normal region is elevated considerably from the Marcus' parabola, which is called the anomalous normal region. Those anomalous energy gap laws are due to the inelastic tunneling mechanism which works actively at the energy gap far from zero. We presented an empirical formula for easily calculating the non-Condon ET rate, which is usable by many researchers. We provided experimental evidence for the anomalous inverted region which was basically reproduced by the present theory. The present theory was extensively compared with the previous non-Condon theories.     

Application of self-consistent HMO theory to heteroconjugated molecules

Substantial improvement of the self-consistent HMO theory, as recently developed by one of us, is made in the parametrization for heterobonds. The theory is extensively applied to a large scale of heteroconjugated molecules including rather complicated ones which are biologically important. The calculated molecular geometries and the wavelengths of optical absorption spectra are found to be in good agreement with experimental values. Examining the effect of inclusion of the -technique to this theory, we find that the calculated values of molecular geometries, electronic spectra and ionization potentials are little affected for most molecules. Dependences of the absorption wavelength and the adiabatic potential on the molecular geometry are also investigated.     

Molecular mechanics of retinal: Twisting mechanism of chromophore in visual pigment

In a model calculation pulli 11-cis retinal (9-cis retinal) from both end sides, the 11–12 double bond (9–10 double bond) is found to be selectively twisted. This property is promising for the twisting mechanism of retinal chromophore in visual pigment as assumed by the Kakitani and Kakitani torsion model.     

Conformationally averaged score functions for electronic propagation in proteins

We explore the influence of conformational dynamics on protein-mediated electron donor-acceptor interactions. We introduce a thermally averaged score function to characterize electronic propagation from redox cofactors into the protein and solvent. The score function is explored for myoglobin at the extended-Hückel level, and the results are compared with those of simpler models. The conformationally averaged quantum results are consistent with the empirical analysis of the Pathways model. Notably, subtle effects of quantum interference among multiple coupling pathways that arise in static structures are largely averaged out when protein thermal motion is included. Propagation through bulk water near the single-protein interface decays rapidly with distance.     

Average electron tunneling route of the electron transfer in protein media

We present a new theoretical method to determine and visualize the average tunneling route of the electron transfer (ET) in protein media. In this, we properly took into account the fluctuation of the tunneling currents and the quantum-interference effect. The route was correlated with the electronic factor in the case of ET by the elastic tunneling mechanism. We expanded by the interatomic tunneling currents 's. Incorporating the quantum-interference effect into the mean-square interatomic tunneling currents, denoted as , we could express as a sum of variant Planck's over 2pi(2). Drawing the distribution of on the protein structure, we obtain the map which visually represents which parts of bonds and spaces most significantly contribute to . We applied this method to the ET from the bacteriopheophytin anion to the primary quinone in the bacterial photosynthetic reaction center of Rhodobacter sphaeroides. We obtained 's by a combined method of molecular dynamics simulations and quantum chemical calculations. In calculating , we found that much destructive interference works among the interatomic tunneling currents even after taking the average. We drew the map by a pipe model where atoms a and b are connected by a pipe with width proportional to the magnitude of . We found that two groups of 's, which are mutually coupled with high correlation in each group, have broad pipes and form the average tunneling routes, called Trp route and Met route. Each of the two average tunneling routes is composed of a few major pathways in the Pathways model which are fused at considerable part to each other. We also analyzed the average tunneling route for the ET by the inelastic tunneling mechanism.     

Spectral tuning of photoactive yellow protein

We report a theoretical study on the optical properties of a small, water-soluble photosensory receptor, photoactive yellow protein (PYP). A hierarchical ab initio molecular orbital calculation accurately evaluated the optical absorption maximum of the wild-type, as well as the lambda(max) values of 12 mutants. Electronic excitation of the chromophore directly affects the electronic state of nearby atoms in the protein environment. This effect is explicitly considered in the present study. Furthermore, the spectral tuning mechanism of PYP was investigated at the atomic level. The static disorder of a protein molecule is intimately related to the complex nature of its energy landscape. By using molecular dynamics simulation and quantum mechanical structure optimization, we obtained multiple minimum energy conformations of PYP. The statistical distribution of electronic excitation energies of these minima was compared with the hole-burning experiment (Masciangioli, T. [2000] Photochem. Photobiol. 72, 639), a direct observation of the distribution of excitation energies.     

An empirical relation between oscillator strengths calculated from the dipole length and dipole velocity formalisms in the optical absorption of conjugated molecules

It is found that a ratio between the oscillator strengths of the optical absorption calculated from the dipole length formalism and those calculated from the dipole velocity formalism is almost constant for many conjugated molecules if the calculation is made using the theoretically obtained transition energy. The value of the ratio becomes very sensitive to the molecular geometry if the calculation is made using the experimentally obtained transition energy. The origin of the constancy of the ratio is discussed.     

Interference, fluctuation, and alternation of electron tunneling in protein media. 1. Two tunneling routes in photosynthetic reaction center alternate due to thermal fluctuation of protein conformation

Electron tunneling routes for the electron transfer from the bacteriopheophytin anion to the primary quinone in the bacterial photosynthetic reaction center of Rhodobactor sphaeroides are investigated by a combined method of molecular dynamics simulations for the protein conformation fluctuation and quantum chemical calculations for the electronic states of the donor, acceptor, and protein medium. The analysis of the tunneling route is made by mapping interatomic electron tunneling currents for each protein conformation. We found that there are two dominant routes mainly passing through Trp(M252) (Trp route) or mainly passing through Met(M218) (Met route). Actual electron tunneling pathways alternate between the two routes, depending on the protein conformation which varies with time. When either the Trp route or the Met route dominates, the electron tunneling matrix element /T(DA)/ becomes large. When both the Trp route and the Met route dominate, /T(DA)/ becomes very small due to the destructive interference of the electron tunneling currents between the two routes. We found that a linear relationship exists between the value of /T(DA)/ and the inverse of the degree of destructive interference Q for a wide range of values (ca. 3-10(3) for Q). A similar relationship was also found previously for electron transfer in ruthenium-modified azurins, suggesting that this relationship holds true in general. From these results, we are led to the conclusion that /T(DA)/ cannot exceed a maximum value at Q = 1, even if much variation of /T(DA)/ happens due to the fluctuation of protein conformation. We also conclude that the property of the electron transfer alternates between constructive and destructive interference, due to the fluctuation of protein conformation. It is impossible to keep a system in either constructive or destructive interference because thermal fluctuation of protein conformation takes place.     

THEORETICAL STUDY OF COLOR CONTROL MECHANISM IN RETINAL PROTEINS. I. ROLE OF THE TRYPTOPHAN RESIDUE, TYROSINE RESIDUE and WATER MOLECULE

Abstract –We calculated the opsin shift due to the electrostatic interaction between tryptophan or tyrosine residues and the chromophore by the perturbation method for various mutual configurations. The obtained opsin shift maps for these configurations demonstrated that when the above residues reside around the ionone ring side, the positive opsin shift (bathochromic shift) is obtained, and when they reside around the Schiff-base side, the negative opsin shift (hypsochromic shift) is obtained. These properties hold true, irrespective of the orientation of those residues, indicating that higher order multipoles of the group play a central role. The maximum value of the opsin shift by these groups amounts to several hundred wavenumbers. These results indicate that the location of some of those amino acid residues at proper positions around the chromophore can cause a considerable opsin shift. We also calculated opsin shift maps for the various mutual configurations between a water molecule and the chromophore for comparison.     

Advanced theory of excitation energy transfer in dimers

Previously, we developed a unified theory of the excitation energy transfer (EET) in dimers, which is applicable to all of the cases of excitonic coupling strength (Kimura, A.; Kakitani, T.; Yamato, T. J. Phys. Chem. B 2000, 104, 9276). This theory was formulated only for the forward reaction of the EET. In the present paper, we advanced this theory so that it might include the backward reaction of the EET as well as the forward reaction. This new theory is formulated on the basis of the generalized master equation (GME), without using physically unclear assumptions. Comparing the present result with the previous one, we find that the excitonic coupling strengths of criteria between exciton and partial exciton and between hot transfer and hopping (F?rster) mechanisms are reduced by a factor of 2. The critical coherency eta c is also reduced significantly.